A kernel \(\kappa \)  is said to be positive definite if the energy \(\kappa (\mu ,\mu )\)  is \({}\geqslant 0\)  for all \(\mu \in \mathfrak {M}\)  (whenever it is well defined); then \(\mathcal {E}\)  is a pre-Hilbert space with the inner product \(\langle \mu ,\nu \rangle :=\kappa (\mu ,\nu )\)  and the energy seminorm \(\Vert \mu \Vert :=\sqrt{\kappa (\mu ,\mu )}\)  [1]}.Any positive definite kernel is of course pseudo-positive, but not the other way around. Except for Sects. REF  and , in this study we shall mainly focus on positive definite kernels, thereby enjoying the advantages of a pre-Hilbert structure on the space \(\mathcal {E}\) . The topology on \(\mathcal {E}\)  determined by this seminorm is said to be strong. A positive definite kernel is said to satisfy the energy principle (or to be strictly positive definite) if the seminorm \(\Vert \cdot \Vert \)  is a norm. (Concerning various potential-theoretic principles, see e.g. the comprehensive work by Ohtsuka [2]}.)
